Examples of Uniform Spaces

Question

Are there any important examples of uniform spaces other than metric spaces and topological groups?

Also, what is an example of when the uniform structure of topological groups is used?

Answer 1

A simple example: $[0,1]^I$ is a uniform space (as a product of metric, hence uniform, spaces). And it's not metrisable if $I$ is uncountable and it's not a topological group as it has the FPP.

In fact, any Tychonov space that is non-metrisable and non-homogeneous is an example.

The uniform structure on topological groups is used e.g. for studying completions, and Haar measure as well.

Answer 2

Examples arbitrary commact Hausdorff spaces, that have a unique uniformity. This example is very interesting, in that it allows to talk about uniform continuity for maps $X\to C$, where, say $X$ is metric and $C$ is compact.

One way uniform spaces are interesting is in the idea of Cauchy completion, which simply generalizes the one for metric spaces. This allows to associate to a given uniform space another one that has nice properties; and it can help in studying, e.g. the topological dynamics of a certain group